Kinetic theory of age-structured stochastic birth-death processes

Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new, fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an aging population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a Bogoliubov-–Born–-Green–-Kirkwood-–Yvon-like hierarchy. Explicit solutions are derived in three limits: no birth, no death, and steady state. These are then compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution

Cladoceran birth and death rates estimates

I. Birth and death rates of natural cladoceran populations cannot be measured directly. Estimates of these population parameters must be calculated using methods that make assumptions about the form of population growth. These methods generally assume that the population has a stable age distribution. 2. To assess the effect of variable age distributions, we tested six egg ratio methods for estimating birth and death rates with data from thirty-seven laboratory populations of Daphnia pulicaria. The populations were grown under constant conditions, but the initial age distributions and egg ratios of the populations varied. Actual death rates were virtually zero, so the difference between the estimated and actual death rates measured the error in both birth and death rate estimates. 3. The results demonstrate that unstable population structures may produce large errors in the birth and death rates estimated by any of these methods. Among the methods tested, Taylor and Slatkin's formula and Paloheimo's formula were most reliable for the experimental data. 4. Further analyses of three of the methods were made using computer simulations of growth of age-structured populations with initially unstable age distributions. These analyses show that the time interval between sampling strongly influences the reliability of birth and death rate estimates. At a sampling interval of 2.5 days (equal to the duration of the egg stage), Paloheimo's formula was most accurate. At longer intervals (7.5–10 days), Taylor and Slatkin's formula which includes information on population structure was most accurate

Inferring transient dynamics of human populations from matrix non-normality

This is the final version of the article. Available from Springer Verlag via the DOI in this record.In our increasingly unstable and unpredictable world, population dynamics rarely settle uniformly to long-term behaviour. However, projecting period-by-period through the preceding fluctuations is more data-intensive and analytically involved than evaluating at equilibrium. To efficiently model populations and best inform policy, we require pragmatic suggestions as to when it is necessary to incorporate short-term transient dynamics and their effect on eventual projected population size. To estimate this need for matrix population modelling, we adopt a linear algebraic quantity known as non-normality. Matrix non-normality is distinct from normality in the Gaussian sense, and indicates the amplificatory potential of the population projection matrix given a particular population vector. In this paper, we compare and contrast three well-regarded metrics of non-normality, which were calculated for over 1000 age-structured human population projection matrices from 42 European countries in the period 1960 to 2014. Non-normality increased over time, mirroring the indices of transient dynamics that peaked around the millennium. By standardising the matrices to focus on transient dynamics and not changes in the asymptotic growth rate, we show that the damping ratio is an uninformative predictor of whether a population is prone to transient booms or busts in its size. These analyses suggest that population ecology approaches to inferring transient dynamics have too often relied on suboptimal analytical tools focussed on an initial population vector rather than the capacity of the life cycle to amplify or dampen transient fluctuations. Finally, we introduce the engineering technique of pseudospectra analysis to population ecology, which, like matrix non-normality, provides a more complete description of the transient fluctuations than the damping ratio. Pseudospectra analysis could further support non-normality assessment to enable a greater understanding of when we might expect transient phases to impact eventual population dynamics.This work was funded by Wellcome Trust New Investigator 103780 to TE, who is also funded by NERC Fellowship NE/J018163/1. JB gratefully acknowledges the ESRC Centre for Population Change ES/K007394/1

Monitoring international migration flows in Europe. Towards a statistical data base combining data from different sources

The paper reviews techniques developed in demography, geography and statistics that are useful for bridging the gap between available data on international migration flows and the information required for policy making and research. The basic idea of the paper is as follows: to establish a coherent and consistent data base that contains sufficiently detailed, up-to-date and accurate information, data from several sources should be combined. That raises issues of definition and measurement, and of how to combine data from different origins properly. The issues may be tackled more easily if the statistics that are being compiled are viewed as different outcomes or manifestations of underlying stochastic processes governing migration. The link between the processes and their outcomes is described by models, the parameters of which must be estimated from the available data. That may be done within the context of socio-demographic accounting. The paper discusses the experience of the U.S. Bureau of the Census in combining migration data from several sources. It also summarizes the many efforts in Europe to establish a coherent and consistent data base on international migration. The paper was written at IIASA. It is part of the Migration Estimation Study, which is a collaborative IIASA-University of Groningen project, funded by the Netherlands Organization for Scientific Research (NWO). The project aims at developing techniques to obtain improved estimates of international migration flows by country of origin and country of destination

Finite-difference approximations and entropy conditions for shocks

Weak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solution. The question arises whether finite-difference approximations converge to this particular solution. It is shown in this paper that, in the case of a single conservation law, monotone schemes, when convergent, always converge to the physically relevant solution. Numerical examples show that this is not always the case with nonmonotone schemes, such as the Lax--Wendroff scheme. 4 figures, 2 tables. (auth

A pace and shape perspective on fertility

Ageing is ubiquitous to all organisms, but ageing does not always mean senescence. Counter to most evolutionary theories of ageing, the patterns of mortality and reproduction may remain unchanged or improve with age, as well as deteriorate. Describing this diversity presents a challenge to eco‐evolutionary demography. The pace–shape framework of mortality tackled this challenge to qualify and quantify orthogonal components of ageing patterns in mortality. Here, we extend this framework to fertility. Analogous to the logic of the mortality framework, we define a perspective, a framework and novel methods for the pace and shape of fertility. These distinguish between orthogonal components of time‐scale (pace) and distribution (shape) of reproduction over adult life span. Our pace and shape framework mirrors that of mortality, through a shift of perspective from the mother giving birth, to the offspring being born. Our new measures overcome many problems associated with measuring natural fertility trajectories, have both a clear biological and mathematical interpretation, can be intuitively visualized and satisfy and extend important conditions of the pace–shape paradigm. A comprehensive framework of fertility pace–shape facilitates ecological and evolutionary research addressing interactions and trade‐offs between components of birth and death patterns, across the whole tree of life. The burgeoning emergence of large comparative demographic data sources across wide environmental, geographical, temporal and phylogenetic ranges, combined with pace–shape measures, opens the door to comparative analyses of ageing which were never possible before

All grown up? The fate after 15 years of a quarter of a million UK firms born in 1998

The theory of firm growth is in a rather unsatisfactory state. However, the analysis of large firm-level datasets which have become available in recent years allows us to begin building an evidence base which can, in turn, be used to underpin the development of more satisfactory theory. Here we study the 239 thousand UK private sector firms born in 1998 over their first 15 years of life. A first, and quite striking, finding is the extraordinary force of mortality. By age 15, 90% of the UK firms born in 1998 are dead, and, for those surviving to age 15, the hazard of death is still about 10% a year. The chance of death is related to the size and growth of firms in an interesting way. Whilst the hazard rate after 15 years is largely independent of size at birth, it is strongly affected by the current (age 14) size. In particular, firms with more than five employees are half as likely to die in the next year as firms with less than five employees. A second important finding is that most firms, even those which survive to age 15, do not grow very much. By age 15 more than half the 26,000 survivors still have less than five jobs. In other words, the growth paths – what we call the ‘growth trajectories’ – of most of the 26,000 survivors are pretty flat. However, of the firms that do grow, firms born smaller grow faster than those born larger. Another striking finding is that growth is heavily concentrated in the first five years. Whilst growth does continue, even up to age 15, each year after age five it involves only a relatively small proportion of firms. Finally, there are two groups of survivors which contribute importantly to job creation. Some are those born relatively large (with more than 20 jobs) although their growth rate is quite modest. More striking though, is a very small group of firms born very small with less than five jobs (about 5% of all survivors) which contribute a substantial proportion (more than one third) of the jobs added to the cohort total by age 15

The Impact of Realistic Age Structure in Simple Models of Tuberculosis Transmission

Background : Mathematical models of tuberculosis (TB) transmission have been used to characterize disease dynamics, investigate the potential effects of public health interventions, and prioritize control measures. While previous work has addressed the mathematical description of TB natural history, the impact of demography on the behaviour of TB models has not been assessed. Methods : A simple model of TB transmission, with alternative assumptions about survivorship, is used to explore the effect of age structure on the prevalence of infection, disease, basic reproductive ratio and the projected impact of control interventions. We focus our analytic arguments on the differences between constant and exponentially distributed lifespans and use an individual-based model to investigate the range of behaviour arising from realistic distributions of survivorship. Results : The choice of age structure and natural (non-disease related) mortality strongly affects steady-state dynamics, parameter estimation and predictions about the effectiveness of control interventions. Since most individuals infected with TB develop an asymptomatic latent infection and never progress to active disease, we find that assuming a constant mortality rate results in a larger reproductive ratio and an overestimation of the effort required for disease control in comparison to using more realistic age-specific mortality rates. Conclusions : Demographic modelling assumptions should be considered in the interpretation of models of chronic infectious diseases such as TB. For simple models, we find that assuming constant lifetimes, rather than exponential lifetimes, produces dynamics more representative of models with realistic age structure

The Leverage of Demographic Dynamics on Carbon Dioxide Emissions: Does Age Structure Matter?

This article provides a methodological contribution to the study of the effect of changes in population age structure on carbon dioxide (CO2) emissions. First, I propose a generalization of the IPAT equation to a multisector economy with an age-structured population and discuss the insights that can be obtained in the context of stable population theory. Second, I suggest a statistical model of household consumption as a function of household size and age structure to quantitatively evaluate the extent of economies of scale in consumption of energy-intensive goods, and to estimate age-specific profiles of consumption of energy-intensive goods and of CO2 emissions. Third, I offer an illustration of the methodologies using data for the United States. The analysis shows that per-capita CO2 emissions increase with age until the individual is in his or her 60s, and then emissions tend to decrease. Holding everything else constant, the expected change in U.S. population age distribution during the next four decades is likely to have a small, but noticeable, positive impact on CO2 emissions

A Multigenerational View of Inequality

The study of intergenerational mobility and most population research are governed by a two-generation (parent-to-offspring) view of intergenerational influence, to the neglect of the effects of grandparents and other ancestors and nonresident contemporary kin. While appropriate for some populations in some periods, this perspective may omit important sources of intergenerational continuity of family-based social inequality. Social institutions, which transcend individual lives, help support multigenerational influence, particularly at the extreme top and bottom of the social hierarchy, but to some extent in the middle as well. Multigenerational influence also works through demographic processes because families influence subsequent generations through differential fertility and survival, migration, and marriage patterns, as well as through direct transmission of socioeconomic rewards, statuses, and positions. Future research should attend more closely to multigenerational effects; to the tandem nature of demographic and socioeconomic reproduction; and to data, measures, and models that transcend coresident nuclear families